Int J Thermophys (2012) 33:105–120DOI 10.1007/s10765-011-1112-x

Frequency Methods Applied to the Characterizationof the Thermophysical Properties of a GranularMaterial with a Cylindrical Probe

Olivier Carpentier · Didier Defer ·Emmanuel Antczak · Thierry Chartier

Received: 5 November 2010 / Accepted: 13 October 2011 / Published online: 30 October 2011© Springer Science+Business Media, LLC 2011

Abstract In many fields, such as in the agri-food industry or in the building industry,it is important to be able to monitor the thermophysical properties of granular materials.Regular thermal probes allow for the determination of one or several thermophysicalfactors. The success of the method used depends in part on the nature of the signal sent,on the type of physical model applied and eventually on the type of probe used andits implantation in the material. Although efficacious for most applications, regularthermal probes do present some limitations. It is the case, for example, when one hasto know precisely the thermal contact resistance or the nature of the signal sent. In thisarticle is presented a characterization method based on thermal impedance formalism.This method allows for the determination of the thermal conductivity, the thermaldiffusivity, and the contact thermal resistance in one single test. The application ofthis method requires the use of a specific probe developed to enable measurement ofheat flux and temperature at the interface of the probe and the studied material. Itspractical application is presented for dry sand.

Keywords Instrumentation development · Intrinsic parameter characterization ·Inverse analysis · Thermal impedance

List of Symbols

a Thermal diffusivity, m2 · s−1

b Thermal effusivity, J · K−1 · m−2 · s−1/2

C Sensor thermal capacity, J · kg−1 · K−1

O. Carpentier (B) · D. Defer · E. Antczak · T. ChartierLaboratoire de Génie-Civil et Géo-Environnement, Université d’Artois,Technoparc Futura, 62400 Béthune, Francee-mail: olivier.carpentier@univ-artois.fr

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f Frequency, Hzl Probe height, mr Probe radius, mR Thermal resistance, K · m2 · W−1

t Time, sZ Thermal impedance, K · m2 · W−1

λ Thermal conductivity, W · m−1 · K−1

φ Fourier transform of heat flux, W · m−2 · Hz−1

θ Fourier transform of temperature, K · Hz−1

ω Pulsation, rad · s−1

Indicesc Contactcpd Computedi Inputo Outputs Sensorth Theoretical∞ Semi-infinite medium

1 Introduction

The in situ monitoring of the thermophysical properties of granular materials is impor-tant in various fields such as the agri-food industry [1–10], pharmacology and biol-ogy [11–15], geology and geothermal energy industry [16–18] as well as in the buildingindustry [19,20]. The thermal conductivity λ, for example, is an important physicalproperty to know when calculating the energy performances of earth-sheltered struc-tures. Thermal diffusivity values are also used in paleoclimatology when carrying outpaleothermal studies to analyze climate changes that occurred in the past [21,22]. Asfor the thermal effusivity b, it allows, among other things, for the monitoring of theevolution of a particular type of soil in terms of water content [23], particularly usefulwhen studying the evolution of the biological, chemical, and physical processes atwork in the soils.

The thermophysical properties of a granular material change over time. Thesechanges can depend on the manufacturing process (raw materials), on their condition-ing and also on their storage (variations in compactness/density and in water content).Natural cycles (frost–thaw alternations) and geological accidents also cause somemodifications in the granular matrix and therefore in the thermophysical properties ofthe granular material.

In order to characterize those materials, thermal probes are being used. Their func-tioning principle relies on the measurement of the evolution of the probe’s temperaturewhen subjected to thermal input. The inverse analysis of the response obtained in termsof the temperature enables the characterization of one or several thermophysical prop-erties [24]. The success of this analysis depends on the nature of the signal sent, on

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the numerical model used as well as on the type of probe used and its implantation inthe material.

Although widely used, those methods can present, in some cases, certain limita-tions. For example, a non-perfect contact between the probe and the material entailsthe presence of a thermal contact resistance Rc. Regarding low values of the contactresistance (Rc < 1 × 10−5 K · m2 · W−1 with standard thermophysical properties ofsoil), the identification of the thermophysical parameters does not require a numericalmodel integrating this particular parameter to be applied. For more significant values(Rc > 1 × 10−4 K · m2 · W−1), a partial knowledge of this parameter does not allowfor a precise identification [25]. (The bigger the grains of the material, the greateris the thermal contact resistance.) The calculation of this resistance shows that it isdamaging to underestimate its influence, regarding dry sands, for example.

In order to solve the contact resistance problem, a non-contact method can be used.Photothermal radiometry [26–32] is a way to determine thermophysical properties ofsamples (plane, cylindrical, or even spherical) for homogeneous, inhomogeneous, andcomposite structures. This non-contact method is based on the study of the responseof a modulated thermal signal at the surface of a sample. The data processing is carriedout in the frequency domain. It is a powerful tool for thermal characterization of broadclasses of materials.

Another important element is the hypothesis formulated concerning the form ofthe signal corresponding to the thermal energy sent into the material. In most physicalmodels (single rod probes), the form of the signal is assumed. The resolution of theinverse problem requires a simple signal (step, ramp) in order to give an analyticalformulation of the indicial response. By studying the asymptomatic behavior of theshort- or long-time analytical formulation, it is possible to simplify the expression andto make it easier to reveal the thermophysical parameters to be identified. The inertiaof the probe then plays an important role. The greater is the inertia of the probe, thegreater is the difference between the form of the assumed signal and the form that isactually injected, thus reducing the accuracy of the identification [25].

The use of single rod probes is adapted to cases where the Biot number is smallwith an inertia contrast close to 1 [25]. In any other circumstances, dual probes andthree-rod probes [33] make it possible to avoid mistakes caused by the uncertaintiesregarding the value of the contact resistance and to the inaccurate form of the excitationsignal.

However, the more rods are being used, the more the granular skeleton and thethermophysical properties of the studied medium are altered, the probes being madeof various elements that need to remain parallel when implanted. This operation canbe difficult with certain types of materials (hard soils, etc.).

Another approach, which has now been developed for several years, consists indissipating a given energy by a plane resistance placed in the material to be charac-terized. A flux and temperature sensor is placed against this resistance allowing us tosimultaneously measure the temperature of the isothermal surface and the flux goingthrough it. The measures are then used in the frequency domain, and from the thermalimpedance, the thermal effusivity of the material can then be derived. The contactresistance between the probe and the material is integrated in the physical model.Previous studies have shown that it could be identified in every test carried out [34].

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In this case, the data processing is carried out by means of a measured flux. Besides,if a temperature sensor is inserted at a known distance from the probe, it is then pos-sible to determine the thermophysical properties of the material as a whole [35]. Thequality of the estimates is then directly linked to the precision of the positioning ofthe temperature sensor. It also has to be noted that it is difficult to set-up a plane rodin situ.

The objective of the probe presented in this experiment is to combine the advantagesof both approaches. On the one hand, the cylindrical geometry of the probe providesus with an easy-to-do setup and the radial diffusion of the heat depends on two of thematerial’s thermophysical parameters (a and λ, for example). On the other hand, thesimultaneous measurement of the flux and of the temperature makes it possible not tohypothesize on the dissipated flux. The contact resistance is identified during the dataprocessing phase based on the use of the thermal impedance formalism. The first testshave been carried out on dry sand.

2 Theoretical Aspects

2.1 Quadrupoles and Thermal Impedance

Analytical methods using thermal quadrupoles are based on the analogy existingbetween the thermal and electrical processes described in the frequency domain (orin the Laplace domain) [36]. Temperature endorses the role of tension and flux therole of intensity. In this formalism, some matrices link the vectors 〈θ, φ〉 as defined inthe isothermal surfaces. θ and φ represent, respectively, the Fourier transformations ofboth temperature and heat flux. Depending on geometries, those surfaces can be plane,cylindrical, or even spherical. In a homogeneous material, two vectors are connectedas follows: (

θiφi

)=

[A BC D

] (θoφo

)(1)

The terms A, B, C , and D of the matrix are defined based on the thermophysi-cal properties of the material and on the distance between the surfaces. The i and oindexes are, respectively, associated to the surface of entry and the system’s output.When various elements (materials, resistance of contact, etc.) are serialized, a globalmatrix is obtained by calculating the product of the elementary matrices:

[AT BTCT DT

]=

n∏k=1

[Ak Bk

Ck Dk

](2)

This formalism is not directly applicable in our approach. The flux density andtemperature are measured for a single isothermal surface. If a relationship betweenthe thermal units at the output of the kind θo = Zoφo is known, we can then write therelationship [34] in the following way:(

θiφi

)=

[AT BTCT DT

] (Zoφoφo

)(3)

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Fig. 1 Schematic representation of the tri-layer and of its characteristic values

We can therefore define the input thermal impedance as follows:

Z i = θi

φi= AT Zo + Bo

CT Zo + DT(4)

2.2 Input Impedance of the Sensor-Material System

The probe developed for this research is described in detail in Sect. 4.2. It is conceivedto promote heat diffusion following axial symmetry. Figure 1 schematically representsthe cylindrical thermal probe apparatus/material to be characterized.

The dotted line (Fig. 1) represents the isothermal surface of measurement of thetemperature and of the flux density. The thermal impedance as defined in this sur-face depends on the thermal characteristics of the different layers penetrated by heatbeyond this surface:

(a) Part of the sensor beyond the isothermal surface. It is a very thin layer and itcan be modeled by localized characteristics such as a thermal resistance and athermal capacity Rs and C . The associated matrix is then expressed as follows:

[1 RsC 1

](5)

(b) Contact resistance probe/material. It is modeled by a pure thermal resistance. Itsmatrix is expressed as follows:

[1 Rc0 0

](6)

(c) Material to be characterized. The frequencies used for measurements are highenough for the thickness of the material not to come into account (Sect. 4.2.1).It behaves as a semi-infinite medium. In its access surface, the temperature and

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the flux in the frequency domain are connected by the thermal impedance char-acteristic of the medium in cylindrical coordinates [37].

Z∞ = 1

2πlλ

K0(ξ)

ξ K1(ξ)(7)

with ξ = r√

jωa

l is the height of the probe and r is the radius of the material’s surface of entry.Kn(ξ), respectively, represent the modified Bessel’s function of the second kind,of n order and ξ argument.

In accordance with the approach described in Sect. 2.1, the input impedance inthe surface of measurement can be calculated.

Z i = K0(ξ) + 2πlλξ K1(ξ)[Rc + Rs]K0(ξ) jCω + 2πlλξ K1(ξ)

[1 + jCω(Rc + Rs)

(1 − Rs

Rc+Rs

)] (8)

In this expression, Rc and C are constant parameters depending on the manufac-turing of the sensor.

2.3 Bessel Functions with Complex Arguments

During the data processing phase, the theoretical impedance will have to be calculated.It includes a Bessel function with complex arguments. The numerical implementationof the calculation of those functions usually yields inaccurate results. However, regard-ing small arguments (ξ < 14), studies have shown [38] that a simplified expression canbe used and still give satisfactory results. In the range of studies concerning our partic-ular media, i.e., for materials with a thermal diffusivity larger than 1 × 10−7 m2 · s−1,a probe radius smaller than 2.5 cm and a measuring time of approximately 15 min,the modulus has maximum values of around 10. The algorithms used in our programto process the data are given below.

Given Kn(ξ) for any argument of the complex number β = Arg(ξ) ∈ {0; π

2

}we

can derive

Kn(ξ) = −(−1)n(

ln

[ | ξ |2

]+ γ + jβ

)In + Ca

n + Cbn (9)

with γ = 0.5572156649. . . as Euler’s constant

Can = 1

2

(−ξ

2

)n

×30∑

k=0

(1

k!(n + k)!(

ξ

2

)2k[

k∑i=1

1

j+

n+k∑i=1

1

j

])(10)

Cbn = 1

2

(ξ

2

)−n

×n−1∑k=0

((−1)k(n − k − 1)!

k!(

ξ

2

)2k)

(11)

with In Bessel’s function of the first kind.

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3 Sensitivity Studies

3.1 Sensitivity to Frequency

The final aim is to proceed to the characterization of some of the system’s thermo-physical properties. This study will be concerned with the thermal diffusivity and thethermal conductivity of the material as well as with the contact resistance. Those prop-erties appear in the expression of the global impedance given by Eq. 8. The resistanceRs and the capacity C of the sensor can be estimated from its constitutive elements. Rshas been estimated at approximately 4 × 10−4 K · m2 · W−1 and C at approximately800 J · kg−1 · K−1. Simulations have shown that resistance Rs could be disregardedin the model. In the identification phase, C has been set at the estimated value. Wehave noticed that even significant variations of C(50 %) have but a limited impact onthe results of the inversion.

The impedance translates the system’s behavior according to the function of thefrequency. Depending on the range under scrutiny, some parameters can play a majorrole while others can be disregarded. The sensitivity study allows for the determinationof the optimal spectral band in order to identify the parameters of interest. Besides,it makes it possible to verify that in the frequency range selected for the study, thereare no correlations between the sensitivities to the parameters and that they can beidentified simultaneously.

For this study, we introduce the impedance sensitivity function Zt to the parameterp by [39]

yp( f ) = p∂ | Zt (Rc, a, λ, f ) |

∂p | Zt (Rc, a, λ, f ) | (12)

The study is limited here to the three unknown parameters a, λ, and Rc.We can see in Fig. 2 that the contact resistance Rc has a non-negligible impact

regarding frequencies higher than 1 × 10−2 Hz. Regarding lower frequencies, theimpedance sensitivity to Rc decreases while the thermophysical properties of a and λ

of the studied medium are predominant. At low frequencies ( f < 1 × 10−3 Hz), theinfluence of the contact resistance is negligible but requires very long testing times. Thechoice of the frequential window [1 × 10−3 Hz; 1 × 10−2 Hz] is a good compromisebetween sensitivity to the parameters sought and testing time duration.

3.2 Correlation Between the Parameters

To carry out a simultaneous identification of a model’s various parameters from aninverse analysis, it is necessary to verify that their respective sensitivities are not cor-related. If such was the case, the optimum set of the parameters would not be unique.The absence of correlation between the parameters is inferred by verifying that eachsensitivity function cannot be obtained by the linear combination of the others. Theabsence of correlation between the parameters taken 2 by 2 can be studied graphicallyby establishing that the two functions are not proportional. If one function is drawn in

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Fig. 2 Sensitivity of inputthermal impedance to thedifferent values identifieddepending on the frequency

Fig. 3 Representation of the correlations between thermal diffusivity, thermal conductivity, and contactresistance

connection with the other, the resulting curve cannot go through the origin. In Fig. 3,we can see that there is no correlation between the contact resistance and the otheridentified parameters. No correlation can be established either between the thermalconductivity and thermal diffusivity. A study has shown that there was no correlationbetween the three parameters.

4 Experimental device

4.1 Test Protocol

In this section, we are presenting the apparatus used to carry out our tests. Details ondata processing are given in the following section. In order to achieve the identificationof the thermophysical properties, the protocol used was the following:

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Fig. 4 Representation of thevarious constitutive layers of thethermal probe

1. Introduction of the probe into the medium and injection of a thermal signal(heating).

2. Measuring the flux and the temperature at the probe/medium interface.3. Calculation of the impedance from the responses in temperature and in flux to the

thermal input.4. Calculation of the theoretical impedance from the physical/numerical models.5. Comparison of the calculated and the theoretical impedance and inverse analysis.6. Identification from the inverse analysis of the thermal conductivity and the diffu-

sivity of the medium as well as its contact resistance.

4.2 Thermal Probe

For the thermophysical characterization of the medium, we are using a cylindricalprobe developed by LGCgE (Fig. 4) The flux density as well as the temperatures arerecorded simultaneously during the test. The bearing of the probe is a hollow vinylpolychloride cylinder of 5 cm diameter. The inside of the probe is insulated withpolyurethane foam. The first constitutive layer of the probe is the heating resistanceequally distributed on the external surface of the bearing. A layer of copper presentingthe same thermophysical properties as the sensor allows us to have an equal dissipationof the heat outwards. Some reservations have been included in order to place threeflux and temperature sensors. Only two sensors are represented in Fig. 4.

This probe prototype has been instrumented with 10 cm high, 1 cm broad, and0.3 mm thick fluxmeters. Each fluxmeter has a 4 µV · W−1 · m−2 sensitivity and atime response of about 150 ms which is fast enough to carry out tests with a time stepof 1 s. The difficulty caused by the bending of the fluxmeter while keeping good sen-sitivity required using a probe of a significant diameter. Thermocouples are integratedto the probe. These are thermocouples of type T with a class one tolerance. Sensorsare connected in a differential setting linked to a HP34970 data processor.

A resistance thermometer (PT100 1/3 DIN) gives the temperature reference. Thecalibration of a fluxmeter is performed in order to determine its sensitivity and to studyits response versus thermal input. Several methods are available [40]. The secondarytransducer method was carried out [41,42]. The advantage of this method residesin the possibility to control thermal equilibrium by means of another uncalibratedfluxmeter [43].

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Fig. 5 Representation of the measuring chain

The fluxmeter to be calibrated is placed on an isothermal cold plate. It is covered bya heat main resistor, an uncalibrated fluxmeter, and an auxiliary resistance, of the samedimensions. On top of this system, an insulating layer is installed in order to reduceheat losses. To ensure perfect contact between these different elements, a heavy solidis placed on the top to press the pile down. The calibration process consists in the mea-surement of the total thermal power provided by the main heat resistor (Joule effect).The provided power is absorbed by the isothermal lower plate. The lost part crossingthe upper fluxmeter is balanced by the second resistor to achieve a zero heat flux.

4.2.1 Test Bench

Tests are carried out in a laboratory. Surrounding conditions (air temperature, hygrom-etry) are stable. The medium is composed of dry sand (water content < 0.5 %) placedin a cylindrical container of 25 cm diameter. A computer-controlled generator regu-lates the thermal resistance by diffusing heat by means of sinusoidal signals (Fig. 5).The probe’s sensors are connected in a differential setting linked to a HP34970 dataprocessor. A subroutine was developed with Labview® to ensure data recording. Resis-tance thermometers (PT100 1/3 DIN) give the temperature reference. The top and bot-tom of the cylindrical container are insulated to guarantee the unidirectional characterof the flux diffusion and prevent humidity exchanges with the surrounding environ-ment. The realization of a series of tests has been automated. An idle period of 6 hhas been observed between every test to achieve stabilized initial conditions. Despitethe low water content of the medium and in order not to cause a hydric transfer, anelevation of 5 ◦C of the temperature at the interface probe-medium will be consideredas maximal.

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Fig. 6 Evolution of the temperature’s amplitude according to distance for a 1 × 10−3 Hz frequency

The theoretical physical model has been established for a transfer regimen undermarginal conditions of a semi-infinite type. To validate this hypothesis, the theoreticalattenuation of the temperature between the entry of the studied medium r (θi) and thetemperature at a given distance r + r∞ (θo) has been calculated (Eq. 13) for a sinu-soidal input at the most unfavorable frequency (i.e., the lowest f = 1 × 10−3 Hz).The rapid damping of the amplitude between input and output temperatures (Fig. 6)shows that from a distance of 3 cm, the thermal input signals are totally damped andthe conditions are indeed those of a semi-infinite type. In our case, the radius of themedium is 10 cm.

∣∣∣∣θo

θi

∣∣∣∣ =∣∣∣∣ Ξ1

Ξ1Ξ2 + Ξ3

∣∣∣∣ (13)

Ξ1 = 1

2πλl

K0(α2)

α2 K1(α2)

Ξ2 = α2[I0(α1)K1(α2) + I1(α2)K0(α1)]Ξ3 = 1

2πλl[I0(α2)K0(α1) − I0(α1)K0(α2) ]

with α1 = r( jω)1/2a−1/2 and α2 = (r + r∞)( jω)1/2a−1/2.In this equation, we refer to Eqs. 1 and 7 and to thermal quadrupoles theoret-

ical analysis [36]. For r = 0, Ξ2 = 1, and Ξ3 = 0, Eq. 13 tends to 1 whenr∞ tends to 0 and tends to 0 when r∞ tends to +∞. However, we have to noticethat the hypothesis of a semi-infinite body is acceptable for common thermophys-ical parameters of civil engineering materials (i.e., 0.5 × 10−7 m2 · s−1 ≤ a ≤1 × 10−7 m2 · s−1, 500 J · K−1 · m−2 · s−1/2 ≤ b ≤ 2000 J · K−1 · m−2 · s−1/2, and

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0.1 W · m−1 · K−1 ≤ λ ≤ 2 W · m−1 · K−1). For other cases, this experiment willneed a new validation of the semi-infinite body hypothesis as regards the lower fre-quency of thermal input.

5 Results and Data Processing

The principle of the data processing consists of experimentally determining the imped-ance of three chosen frequencies. The theoretical physical model of the impedancedepends on three parameters to be identified. Estimating the three parameters is donewith a system of three equations with three unknown variables. From an experimentalpoint of view, the energy dissipated will be defined as a linear combination of threesinusoidal functions.

P(t) =3∑

i=1

Ai sin(2π fi t + ϕi ) (14)

In this relationship, f represents each input frequency, and A and ϕ represent,respectively, the amplitude and the associated phase shift.

During the experimental campaign (26 tests), the frequencies 1/125 Hz, 1/250 Hz,and 1/500 Hz were selected. Between each test, we subjected amplitudes and phaseshifts to variations. Sinusoidal amplitude is selected in a way that the maximal valuecannot be larger than the power of the generator and this is in order to avoid saturationin our input signal. Besides, the medium being granular, a too powerful input couldalter its thermal properties (variation in water content).

The selected frequencies are within the frequency band we have targeted duringour sensitivity study. They are selected because they are distant enough from eachother, making their identification easier for the inverse analysis. Indeed, for the inverseanalysis, we are using a simplex algorithm to minimize a nonlinear function. If thefrequencies were too close to each other, it would lead to the classic situation of theill-posed problem that would not guarantee the uniqueness of its solution, its physicalmeaning, or its stability during the inverse analysis.

To ensure good behavior of our numerical model during the two different stages ofour calculation, we are comparing the temperatures and the flux we calculated withthose we had modeled (Figs. 7, 8).

We then proceed to the numerical calculation of the Fourier transformation of thosesignals (flux and temperature). The Fourier transformation of each signal is limitedto a three-line spectrum. We can therefore deduce the experimental impedance of ourmedium Zcpd for the three frequencies considered.

Zcpd( fi ) = θi ( fi )

φi ( fi )(15)

Then, the theoretical thermal impedance Z th is calculated from the expression(Eq. 8) for the three frequencies from the initial values usually observed for geoma-terials. The identification of the parameters is done by minimizing an error function

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Fig. 7 Comparison of the calculated flux with those modeled

Fig. 8 Comparison of the calculated temperatures with those modeled

(S0) between the experimental and theoretical values of the impedance. A simplexalgorithm was used,

S0 =3∑

i=1

(| Zcpd,i − Z th,i |)2 (16)

Figure 9 represents the results of an optimization. Variations between the theoreticalimpedances Z th( f ) and the impedances calculated Zcpd( f ) for the three frequenciesof solicitation can be observed. Only the modulus is represented. Theoretical andcalculated phases were similar.

Optimization concerns the thermophysical parameters a and λ as well as the contactresistance Rc. In this test, the estimated values are: a = 1.67 × 10−7 m2 · s−1, λ =0.249 W · m−1 · K−1, and Rc = 4.69 × 10−3 K · m2 · W−1.

The values identified with the fluxmetric thermal probe are compared to the refer-ence values obtained with the guarded hot-plate method (ISO 8302), tests carried outin our laboratory. A synthesis of the tests is represented in Fig. 9 and Table 1.

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Fig. 9 Comparison betweentheoretical Zth impedances andcalculated Zcpd impedances forthe characteristic frequencies ofinput signal

Fig. 10 Thermal diffusivity and conductivity identified during the 27 tests carried out with the fluxmetricthermal probe

Table 1 Comparison of the thermophysical properties identified with the probe with those obtained withthe thermal characterization bench

a (m2 · s−1) λ (W · m−1 · K−1) Rc (K · m2 · W−1)

Classicial method 1.89 × 10−7 0.242 –

Cylindrical probe 1.71 × 10−7 0.238 4.7 × 10−3

Error 9.5 % 1.65 % –

The average contact resistance identified during the tests is 4.7×10−3 K · m2 · W−1

and hardly varies. A good correlation is to be noted between the tests carried out atthe thermal characterization bench and the values measured with the probe regardingthe thermal effusivity and diffusivity values. An error in effusivity values will have aninfluence on the other factors. This can be explained by the fact that the diffusivity isaltered by the radius of the probe, a radius which has not been optimized yet. Studies

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are currently being done to improve the geometry of the probe and to make it moresensitive to diffusivity measurements (Fig. 10).

6 Conclusion

In this article, we have shown how to identify in a single test the thermal diffusivity, thethermal conductivity, and the contact resistance of a granular medium from a thermalprobe. The use of frequency methods allows us to focus on the relevant informationsent by the signals which entails an extremely short time of calculation during theinverse analysis. Besides, to be able to identify the contact resistance in every testcarried out makes it a particularly reliable method for in situ measurements. Even-tually, the identifications provided are extremely useful regarding the conductivity.Results are, however, less satisfactory in terms of thermal diffusivity identification,compelling us to achieve better optimization standards of our probe’s instrumentation,optimization we are currently developing in our research department.

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